On the collective properties of a boson system

Abstract

A method in close correspondence with the theory of super-conductivity of Bardeen, Cooper and Schrieffer is applied to a boson system, extending an earlier approach byBogolubov. Minimizing the energy with respect to a trial ground state vector of exponential form, new equations and expressions are derived for the excitation spectrum of bosons. The expressions are in a one to one correspondence with those for fermions, from which formally they differ only through signs. The physical content of the expressions is, however, very different. Equations with a predominantly repulsive interaction which in the fermion case have no collective solutions, lead to such solutions for bosons because of a partial Bose-Einstein condensation. It is pointed out that the method is related to a linearization of the quantized matter field equations. Temperature dependent equations are obtained by the same methods as in the theory of superconductivity. The λ-point is defined by the disappearance of the partial Bose-Einstein condensation.